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Vol. 51, 25–42 ©2022

http://doi.org/10.21711/231766362022/rmc512

Some remarks on an arbitrary-order SIR

model constructed with Mittag-Leﬄer

distribution

Noemi Zeraick Monteiro 1and Sandro Rodrigues

Mazorche 1

1Universidade Federal de Juiz de Fora, Juiz de Fora, Brazil

Abstract. Our recent works discuss the meaning of an arbitrary-

order SIR model. We claim that arbitrary-order derivatives can be

obtained through special power-laws in the infectivity and removal

functions. This work intends to summarize previous ideas and show

new results on a meaningful model constructed with Mittag-Leﬄer

functions. We emphasize the tricky idea to deal with equilibria, the

nonlocality of the model and the non-intuitive behavior near the

lower terminal.

Keywords: Fractional Calculus. Epidemiological model. Mittag-

Leﬄer functions. Nonlocality. Non-intuitive behaviors.

2020 Mathematics Subject Classiﬁcation: 12A34, 67B89.

1 Introduction

“Mathematics is biology’s next microscope, only better; biology is math-

ematics’ next physics, only better” [1]. The quote by biomathematician

The ﬁrst author is supported by the Coordination for the Improvement of Higher

Education Personnel (CAPES) - Brazil - Financing Code 001; e-mail of the correspond-

ing author: nzmonteiro@ice.ufjf.br.

25

26 N. Zeraick Monteiro and S. R. Mazorche

Joel E. Cohen portrays the current development of mathematical biology,

a branch that holds the attention of several researchers and has been on

the agenda every day during the current COVID-19 pandemic. The wide

use of mathematical models in situations related to biology does not ex-

haust their study. Contrariwise, some mathematical tools need a revision

to be used properly.

A great tool for problem modeling is Arbitrary-Order Calculus, known

as Fractional Calculus. In the most used deﬁnitions, there is the possi-

bility of explicitly considering the dependence of previous stages of the

phenomenon studied, through the nonlocality of the operators. This is

generally related to the “memory eﬀect” [2]. The recent explosion of pub-

lications in Fractional Calculus highlights its immense applicability in nu-

merous areas (see, for instance, the data collected in [2]). However, the

basis is still not uniﬁed and coherent.

Compartmental models, for example, have been widely studied with

arbitrary orders. Generally, they are obtained by replacing an integer

derivative with an arbitrary-order one. Throughout our research, we pub-

lish some works about meaning diﬃculty, loss of properties and the lack

of the construction of fractional SIR type models ([3], [4], [5]). In this

context, we study a model proposed by Angstmann, Henry and McGann

[6] in which the arbitrary-order derivatives are obtained by construction,

considering Mittag-Leﬄer functions and generalizing the infectivity and

remotion functions. We seek to extend some analytical and numerical

results of the model in [4], [7], [8], [9].

Here, one of the aims is, after some preliminary presentations, dis-

cuss the idea around how to ﬁnd equilibria in the proposed model, since

equating the right side of the system to zero is no longer a viable strategy.

The discussed strategy is somewhat trickier, using Laplace transform tech-

niques. After that, the main propose deals with two points that were not

discussed yet: as one would expect, the model is nonlocal and, moreover,

presents a non-intuitive behavior in the lower terminal.

Some remarks on an arbitrary-order SIR model 27

2 A brief note about nonlocality and the Frac-

tional Calculus

The impetus theory, studied by names as Leonardo Da Vinci, deals

with the concept of “impression”. According to Da Vinci, the impression is

maintained during a certain time in its sensitive object. This impression

(memory) characterized by a long time produces more lasting eﬀects, while

a short memory produces eﬀects that occur in shorter times [10]. Although

this reasoning was lost in the context of the Newtonian physics, in the last

century the quantum theory and the modern string theory allowed a revival

of nonlocal theories. Within that context, the Fractional Calculus can be

seen as a continuation or a sublimation of nonlocal concepts [11].

Broadly speaking, the Fractional Calculus, parallel to the delay diﬀer-

ential equations, has been shown to be a very useful tool in capturing the

dynamics of the physical process of several scientiﬁc objects. Probably, it

was born in 1695, when l’Hôpital asked Leibniz about the meaning of a

derivative of order 1/2. Over the subsequent centuries, important advances

were made by Liouville, Riemann, Grünwald, Caputo, and many others.

However, it was only after the ﬁrst International Conference on Fractional

Calculus and Applications, in 1974, that the number of researchers in Frac-

tional Calculus showed great growth. The reader may refer to the reference

[12] for a detailed chronology of publications in Fractional Calculus until

2019, as well as for general results.

Nonlocal operators can be constructed in diﬀerent ways, depending on

the bias worked on. Here, before introducing the arbitrary-order integral,

we recall the concept of the integer-order integral, sometimes called the

multiple or iterated integral:

Deﬁnition 2.1 (Integer-order iterated integral).The integral of order

n∈Nis deﬁned by the expression

Inf(t) = Zt

0Zt1

0Zt2

0

· · · Ztn−2

0Ztn−1

0

f(tn)dtndtn−1· · · dt3dt2dt1.(2.1)

By deﬁnition, I0f(t) = f(t).

28 N. Zeraick Monteiro and S. R. Mazorche

The next result, using the Laplace convolution, is a starting point for

the generalization of the concept of an integral of order n. For that, we

deﬁne:

Deﬁnition 2.2 (Gel’fand-Shilov function).Let α∈R,α > 0. The

Gel’fand-Shilov function is deﬁned as

ϕα(t) =

tα−1

Γ(α)if t≥0,

0if t < 0,

(2.2)

where Γrepresents the gamma function.

Theorem 2.3. Let n∈N,0< t < ∞and f(t)be an integrable function.

Then,

Inf(t) = ϕn(t)⋆ f(t) = Zt

0

(t−τ)n−1

(n−1)! f(τ)dτ, (2.3)

where ⋆indicates convolution [13].

Thus, it is to be expected that the deﬁnition of an integral of arbitrary

order αis given by Iαf(t) = ϕα(t)⋆ f(t). Below, we consider [a, b]a ﬁnite

real interval, and αa real number such that 0≤n−1< α < n, with n

integer:

Deﬁnition 2.4 (Riemann-Liouville integral in ﬁnite intervals).The Riemann-

Liouville integral of an arbitrary order αis set to t∈[a, b]by

Iα

a+f(t) = 1

Γ(α)Zt

a

(t−θ)α−1f(θ)dθ. (2.4)

After introducing the arbitrary-order integral, it is natural to search

for the deﬁnition of the corresponding derivative. There are several deﬁ-

nitions of these kind of derivatives, each one constructed with a particular

viewpoint. In this work, we use the Riemann-Liouville’s one:

Deﬁnition 2.5 (Riemann-Liouville derivative in ﬁnite intervals).The

Riemann-Liouville derivative of an arbitrary order αis set to t∈[a, b]

by

Dα

a+f(t) = Dn[In−α

a+f(t)] = 1

Γ(n−α)dn

dtnZt

a

(t−θ)n−α−1f(θ)dθ , (2.5)

Some remarks on an arbitrary-order SIR model 29

with Dnrepresenting the integer-order derivative.

Finally, we present the Mittag-Leﬄer functions with one, two, and

three parameters. The classic Mittag-Leﬄer function, due to its impor-

tance in several arbitrary-order diﬀerential equations, was nicknamed the

“queen of special functions” of the Fractional Calculus. Its importance

for Fractional Calculus is analogous to the signiﬁcance of the exponential

function for classical Calculus. We present the following deﬁnition [12]:

Deﬁnition 2.6 (Mittag-Leﬄer function with one, two, and three param-

eters).Let zbe a complex number, and three parameters α, β complex,

and ρreal, such that Re(α)>0, Re(β)>0, ρ > 0. We deﬁne the Mittag-

Leﬄer function with three parameters through the power series

Eρ

α,β (z) =

∞

X

k=0

(ρ)k

Γ(αk +β)

zk

k!,(2.6)

where (ρ)kis the Pochhammer symbol, deﬁned by (ρ)k= Γ(ρ+k)/Γ(ρ).

The three-parameter Mittag-Leﬄer function is also called Prabhakar

function. Particularly, when ρ= 1, we have (ρ)k=k!. In this case,

the deﬁnition recovers the two-parameter Mittag–Leﬄer function, denoted

simply by E1

α,β(t) = Eα,β (t). When ρ=β= 1, we obtain the classic

Mittag-Leﬄer function, denoted by E1

α,1(t) = Eα,1(t) = Eα(t). Finally, we

recover the exponential function when α=β=ρ= 1.

3 The model

We present in [4] a physical derivation following the steps of Angst-

mann, Henry & McGann [6], which use the probabilistic language of Con-

tinuous Time Random Walks (CTRW), and Mittag-Leﬄer functions. As

we can see with more detail in the references, the ﬁrst idea is to consider

an individual infected since the time t′. If there are S(t)susceptible in

time t, this infected person has a probability S(t)/N that his contact is

susceptible, considering the population homogeneous. Therefore, in the

period of tto t+ ∆T, the expected number of new infections per infected

30 N. Zeraick Monteiro and S. R. Mazorche

individual is given by σ(t, t′)S(t)∆T/N. The transmission rate per infec-

tious individual σ(t, t′)depends on both the age of the infection, t−t′,

and the present time, t. The probability that an individual infected at the

moment t′is still infected at the moment tis given by the survival function

Φ(t, t′). Therefore, the ﬂux of individuals to the Icompartment at a time

tis recursively given by

q+(I, t) = Zt

−∞

σ(t, t′)S(t)

NΦ(t, t′)q+(I, t′)dt′.(3.1)

To deal with the individuals infected at the time 0, we consider the

time in which each individual has become infected. This is given by the

function i(−t′,0) which represents the number of individuals who are still

infectious at time 0and who were originally infected at some point earlier

t′<0. Then, q+(I, t′) = i(−t′,0)/Φ(0, t′)for t′<0. For simplicity, we

consider i(−t, 0) = i0δ(−t), where δ(t)is the Dirac delta function. So,

q+(I, t) = Zt

0

σ(t, t′)S(t)

NΦ(t, t′)q+(I, t′)dt′+i0σ(t, 0)S(t)

NΦ(t, 0).(3.2)

As said, the infection rate σ(t, t′)is assumed to be a function of both

the current time (due, for example, to containment measures), having an

extrinsic infectivity, ω, and the age of infection, t−t′, having an intrinsic

infectivity, ρ. So, we can write

σ(t, t′) = ω(t)ρ(t−t′).(3.3)

Assuming that the natural death and the removal of an infected indi-

vidual are independent processes, we can write the survival function as

Φ(t, t′) = ϕ(t−t′)θ(t, t′),(3.4)

where ϕ(t−t′)is the probability that an individual infected since t′has not

yet recovered or been killed by the disease at time t. Also, θ(t, t′)is the

probability that an infected individual since t′has not yet died of natural

death (that is, independent of the disease) until time t. The θfunction is

given by θ(t, t′) = e−Rt

t′γ(u)du,where γis the death rate.

Some remarks on an arbitrary-order SIR model 31

We deﬁne infectivity and recovery memory kernels

KI(t) = L−1L{ρ(t)ϕ(t)}

L{ϕ(t)}, KR(t) = L−1L{ψ(t)}

L{ϕ(t)},(3.5)

where ψ(t) = −dϕ(t)/dt. This ψhas an important relationship with the

continuous random variable Xthat provides the time of removal of the

individual from the infectious compartment. The cumulative distribution

of X, namely Fdeﬁned by F(t) = P(X≤t), is such that F(t) = 1 −ϕ(t).

Therefore, the probability density function of Xis ψ(t) = −dϕ(t)/dt. We

can state the set of equations for the SIR model in a similar manner to

that written originally by Kermack and McKendrick [14]:

dS(t)

dt =γ(t)N−ω(t)S(t)

Nθ(t, 0) Zt

0

KI(t−t′)I(t′)

θ(t′,0)dt′−γ(t)S(t),(3.6)

dI(t)

dt =ω(t)S(t)

Nθ(t, 0) Zt

0

KI(t−t′)I(t′)

θ(t′,0) dt′−θ(t, 0) Zt

0

KR(t−t′)I(t′)

θ(t′,0) dt′−γ(t)I(t),

(3.7)

dR(t)

dt =θ(t, 0) Zt

0

KR(t−t′)I(t′)

θ(t′,0)dt′−γ(t)R(t),(3.8)

where we consider the same rate γ(t)of natural mortality in each compart-

ment, with the birth rate equal to that. The population remains constant.

We choose ψ(t)and ρ(t)using Mittag-Leﬄer functions, in order to

generalize the exponential distribution of the random variable Xand allow

a variable intrinsic infectivity:

ϕ(t) = Eα,1−t

τα, ρ(t) = 1

ϕ(t)

tβ−1

τβEα,β −t

τα.(3.9)

Using Laplace transform techniques, the Riemann-Liouville derivatives

arise along the construction and the SIR model, with 1≥β≥α > 0, is

given by

dS(t)

dt =γ(t)N−ω(t)S(t)θ(t, 0)

Nτ βD1−βI(t)

θ(t, 0)−γ(t)S(t),(3.10)

dI(t)

dt =ω(t)S(t)θ(t, 0)

Nτ βD1−βI(t)

θ(t, 0) −θ(t, 0)

ταD1−αI(t)

θ(t, 0) −γ(t)I(t),(3.11)

dR(t)

dt =θ(t, 0)

ταD1−αI(t)

θ(t, 0)−γ(t)R(t),(3.12)

32 N. Zeraick Monteiro and S. R. Mazorche

Notice that, if α=β= 1, and γ(t)≡γ , ω(t)≡ωare considered

constant, we get the simple integer-order SIR model with constant coef-

ﬁcients. Moreover, the cumulative distribution of Xis a Mittag-Leﬄer

distribution F(t;α, τ )=1−Eα(−(t/τ)α). If α=β= 1, we have an ex-

ponential distribution and the expectation (ﬁrst moment) of the random

variable Xexists, with τbeing exactly the average recovery time. When

α < 1, we do not have ﬁnite expectation.

Remark 3.1. In epidemics such as COVID-19, reports exhibit the asym-

metry of each infectious wave: “while COVID-19 accelerates very fast, it

decelerates much more slowly. In other words, the way down is much

slower than the way up” [15]. In addition, scientists suggest that infected

people are most infectious immediately before they develop symptoms and

at the onset of them (e.g. [16]). The two factors presented, that is, the

asymmetry of the data, with a heavy right tail eﬀect, and the decrease in

infectivity over time since infection, are captured by the arbitrary-order

model presented. As started in [4], [17], [18], we have been working on

applications of the model to COVID-19.

3.1 Equilibrium

Here, we analyze the equilibrium point (S∗, I∗, R∗)such that lim

t→∞(S, I, R) =

(S∗, I∗, R∗), where the limit is taken coordinate by coordinate. We con-

sider γ(t)≡γconstant, so θ(t, 0) = e−γt. Taking the limit t→ ∞, the

model reduces to:

0 = γN −lim

t→∞ ω(t)S(t)e−γt

Nτ βD1−β(I(t)eγ t)−γS∗,(3.13)

0 = lim

t→∞ ω(t)S(t)e−γt

Nτ βD1−β(I(t)eγ t)−e−γ t

ταD1−α(I(t)eγt )−γI ∗,(3.14)

0 = lim

t→∞ e−γt

ταD1−α(I(t)eγt )−γR∗.(3.15)

To calculate the limits of the form lim

t→∞ e−γt D1−α(I(t)eγt ), considering

γ > 0, we follow [19] and take the Laplace Transform:

L{e−γt D1−α(I(t)eγt )}= (s+γ)1−αL{I}.(3.16)

Some remarks on an arbitrary-order SIR model 33

Using a Taylor series expansion, we get

(s+γ)1−αL{I}=L{I}(γ1−α+ (1 −α)γ−αs+O(s2)).(3.17)

As the Laplace Transform is a linear operator, we can invert term by

term, obtaining

e−γt D1−α(I(t)eγt ) = γ1−αI(t) + (1 −α)γ−αdI

dt +L−1(O(s2)).(3.18)

As we consider lim

t→∞ I(t) = I∗, we have

lim

t→∞ dI/dt = lim

t→∞ L−1(O(s2)) = 0.(3.19)

So, it follows that

lim

t→∞ e−γt D1−α(I(t)eγt ) = γ1−αI∗.(3.20)

Substituting these results into Eq. (3.13)-(3.15) and assuming that

lim

t→∞ ω(t) = ω∗(possibly ω∗= 0), we are left with

0 = γN −ω∗S∗

Nτ βγ1−βI∗−γS∗,(3.21)

0 = ω∗S∗

Nτ βγ1−βI∗−1

ταγ1−αI∗−γI ∗,(3.22)

0 = 1

ταγ1−αI∗−γR∗.(3.23)

These equations make it possible to obtain a disease-free state:

S∗=N, I ∗= 0, R∗= 0,(3.24)

and, in the case where ω∗>0, we also obtain an endemic state:

S∗=((τ γ)β−α+ (τ γ )β)N

ω∗, I∗=N(τ γ )α

1+(τ γ)α−N(τ γ )β

ω∗, R∗=N

1+(τ γ)α−N(τ γ )β−α

ω∗.(3.25)

When ω(t)≡ω, we get ω∗=ωand recover the endemic state of the

original article [6]. We observe that the endemic state makes physical sense

only if we can have I∗>0and R∗>0, that is, if

ω∗>(τγ)β−α+ (τγ)β.(3.26)

34 N. Zeraick Monteiro and S. R. Mazorche

We have thus proven that, if there are asymptotically stable equilibria

in the case γ > 0, then they are given by Eq. (3.24)-(3.25). However,

we have not really proved that these states are asymptotically stable equi-

libria. We expect the disease-free state to be an asymptotically stable

equilibrium when ω∗<(τγ)β−α+ (τ γ)β, while the endemic state must be

asymptotically stable if ω∗>(τγ)β−α+ (τ γ)β.

For the case without vital dynamics, that is, γ= 0, if α=βand ω(t)≡

ω, we can write dS/dR =−(ωS(t)/Nτ), as in the original model, also

discussed in [4]. Thus, the equilibrium point is the same as in the original

case, as we state in [7]. For the case with γ > 0, there are diﬃculties to

analyze formally the stability. Some advances were also reported in [7].

4 Main remarks

There are several numerical methods that can be applied to arbitrary-

order derivatives. For now, we build a numerical L1-scheme [20] to dis-

cretize the model described in the last Section. The time interval [a, t]

is discretized as a=t0< t1<· · · < tn=t, where the time steps

∆Ti=ti+1 −ti, for i∈ {0,· · · , n −1}, have the same size ∆T. Consid-

ering α∈(0,1],we perform the following discretization for the Riemann-

Liouville derivative:

D1−α

a+f(tj)≃∆Tα−1

Γ(α+ 1) j−1

X

k=0

f(tk)[(j−k+ 1)α−2(j−k)α+ (j−k−1)α] + f(tj),(4.1)

where ti=i∆T+t0for all i∈ {0,1,· · · , n}. It is important to state that

the integer-order case is obtained by taking α= 1.

4.1 Nonlocality

We illustrate that the model (3.10)-(3.12) is nonlocal, as expected. For

this, we consider N= 106and initial conditions S(0) = N−1,I(0) =

1,R(0) = 0. At time t= 90, the numerical solution gives S(90) =

205890,I(90) = 267840, and R(90) = 526270. We now consider this

initial condition and run the model again.

Some remarks on an arbitrary-order SIR model 35

The Figure 4.1 illustrates the change of the solution, where the dashed

line corresponds to the solution from the time t= 90. In Figure 4.2,

we have the equivalent trajectories for a maximum time T= 3000. The

equilibrium is maintained.

0 50 100 150 200 250 300

t in days

0

2

4

6

8

10

S (blue), I (red), R(black)

#105

Parameters: !=3; ==10; .=0.01

,=0.7; -=0.9

Figure 4.1: Change in the solution.

0246810

S#105

0

0.5

1

1.5

2

2.5

3

I

#105

Parameters: !=3; ==10; .=0.01

,=0.7; -=0.9

Figure 4.2: Change in the trajectory.

Remark 4.1. In the integer-order SIR model, the epidemiological mean-

ingful parameters deﬁne the epidemic independently of time. Given an

initial condition (S(0), I(0), R(0)), and the solution (S(t), I (t), R(t)) of

the classic SIR model, let t∗>0: if we start at time t∗, with initial condi-

tion (S(t∗), I(t∗), R(t∗)), the solution will continue to be (S(t), I(t), R(t))

for t > t∗. This property of autonomous dynamical systems is called

invariance of solutions [21]. However, this is not valid here, making it

diﬃcult to correctly choose initial conditions: what about the past prior

to the considered initial point? In this model, to adjust to the infection,

the parameters depend on the time series prior to the starting point. In

other words, diﬀerent pasts will lead to diﬀerent solutions in the future.

Starting a modeling in the second month of an epidemic, for example, does

not lead to the same result as when we start on the ﬁrst day. It is worth

to mention that, to obtain the Equation (3.2), we used the Dirac delta

function to indicate that the disease does not exist before the time 0.

The nonlocality also implies an unexpected behavior of the reproduc-

tion number at time t′. It can be understood as the expected number

of individuals that are infected by an infectious individual since t′, with

36 N. Zeraick Monteiro and S. R. Mazorche

the basic reproduction number being the average number of secondary

infections that occur when an infectious individual is introduced into a

completely susceptible population [22]. In the constructed model, it is not

natural to deﬁne a formula that provides the reproduction number. Then,

in [6] the authors propose a construction for the basic reproduction num-

ber through an integral. Here, we extend the proposal of the reproduction

number for any time t′. Thus, initially, we consider ω(t)constant and the

deﬁnitions

ℜ0=ωγα−β

τβγα+τβ−α,(4.2)

ℜ(t′) = S(t′)ω

Nτ β

γα−β

γα+τ−α=ℜ0·S(t′)

N.(4.3)

This analysis is important, since for this model we do not have param-

eter tables, nor even prior application to any speciﬁc infectious disease.

As we can see here, in classical epidemiological models, the peak occurs

when ℜ(t)=1, but this does not occur in models of arbitrary orders or

with time-dependent parameters.

We consider the example below, in Figure 4.3, where the value of ℜ(t′)

given by Eq. (4.3) for the peak point t′is ℜpeak ≈0.9078 <1. If we

start modeling at a time before the peak, for which ℜpeak <ℜ(t)<1,

the epidemic will decrease even faster, but after a small rise. Note that

α=β. This will be discussed later. Also, is also possible to have no more

peaks if we start in a previous point, as we can see in Figure 4.4. In both

examples, we consider N= 106and the initial condition is (N−1,1,0).

We also can observe ℜpeak >1. In some of these cases, we can take an

earlier day, where ℜ(t)>ℜpeak >1, and the disease starts decreasing even

faster, as illustrated in Figure 4.5. In other words, “the point where the

epidemic must start to decrease” is really not the peak point anymore! This

indicates a diﬃculty in understanding the ℜ0deﬁned in (4.2): although

ℜ0>1is the condition of endemic equilibrium viability given in (3.25),

ℜ0S(0)/N > 1does not necessarily indicate that the epidemic will or will

not occur, that is, whether or not Iwill grow.

Some remarks on an arbitrary-order SIR model 37

56 58 60 62 64 66 68 70

t in days

1.36

1.38

1.4

1.42

1.44

1.46

I

#105

Parameters: !=2; ==4; .=0.01

,=0.5; -=0.5

<(p-1)=0.9455

<(p)=0.9290

Figure 4.3: Case ℜ(p)<1(α=β).

100 105 110 115 120 125 130 135

t in days

1.55

1.6

1.65

1.7

1.75

I

#105

Parameters: !=2; ==7; .=0.01

,=0.5; -=0.6

<(p-1)=0.9878

<(p)=0.9750

Figure 4.4: Case ℜ(p)<1.

In this same Figure 4.5, we consider diﬀerent starting points along

the originally constructed Icurve with initial condition (N−1,1,0). So,

it illustrates the evolution of the Icompartment if we start modeling at

diﬀerent starting points along the same curve. The nonlocal behavior of

the model is surprising and we see that, in this example, the Icompartment

only rises if we take a starting point much earlier than the peak. Here, we

also start to consider the proposal that we presented in [7] for a S-variable

reproduction number:

ℜS(t′) = Z∞

t′

S(t)

Nτ βω(t)e−γ(t−t′)(t−t′)β−1Eα,β −(t−t′)

ταdt. (4.4)

In this proposal, the basic reproduction number is given by

ℜS

0=Z∞

0

S(t)

Nτ βω(t)e−γ ttβ−1Eα,β −t

ταdt. (4.5)

In the Figure table 4.6, we display the ℜ0S(0)/N and the ℜS

0S(0)/N

for each curve. In classical models, the disease declines since the beginning

if and only if ℜ0S(0)/N < 1. If ℜ0S(0)/N > 1, it must grow for a while.

As we can see, this does not happen with this model.

However, the equilibrium I∗given in (3.25) is the same for any initial

condition. So, broadly speaking, two simulations of the same disease with

diﬀerent starting points lead to diﬀerent solutions, but to the same end,

what is really important to study, for example, how many people will be

infected in total. Here we pointed out that, as will be explored in next

38 N. Zeraick Monteiro and S. R. Mazorche

25 30 35 40 45 50 55 60

t in days

1

1.5

2

2.5

3

3.5

4

4.5

I

#105

Parameters: !=5; ==9; .=0.01

,=0.5; -=0.9

Figure 4.5: Modifying the start point. Figure 4.6: ℜ0and ℜS

0.

0 100 200 300 400 500

t in days

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

I

#105

Parameters: !=5; ==9; .=0.01

,=0.5; -=0.9

Figure 4.7: The equilibrium remains the same.

works, the S−variable reproduction number seems to provide something

closer to what we expect, in terms of plausible biological arguments.

4.2 Behavior near lower terminal

Finally, we exhibit an unusual feature observed in the initial instants

of the model simulations. One can easily see in the Figure 4.5 that the

Some remarks on an arbitrary-order SIR model 39

blue and purple curves have a small depression before a rise. This type of

behavior does not appear in the integer-order model.

In fact, this can be explained given the asymptotic behavior of the

Riemann-Liouville derivatives near the lower terminal [23]:

Dα

a+f(t)≈f(a)

Γ(1 −α)(t−a)−α,(t→a+).(4.6)

So, in the model (3.10)-(3.12), if β > α, we have dI/dt < 0for tsmall

enough, as illustrated in the Figures 4.8-4.9.

0 0.2 0.4 0.6 0.8 1

t in days

0.7

0.8

0.9

1

1.1

1.2

1.3

I

Parameters: !=3; ==10; .=0.01

,=0.2:0.1:0.9; -=0.9

Figure 4.8: Lower terminal - Ex.1.

0 0.1 0.2 0.3 0.4

t in days

1

1.2

1.4

1.6

1.8

2

2.2

I

Parameters: !=3; ==10; .=0.01

,=0.4; -=0.4:0.1:0.9

Figure 4.9: Lower terminal - Ex.2.

When α=β, this behavior is not observed. For instance, we can see

in the pink curve of the Figure 4.3 that the behavior in the lower terminal

is the traditional movement of rising to a peak and decreasing after that.

Here, we also emphasize that the model assumes that there was no

disease before time 0, and then, at that point, an impulse caused the

disease to start. So, the historic is theoretically considered completely,

avoiding dubiousness about the starting point. However, the solution is not

C1, as it is not derivable at the origin. These remarks aim to strengthening

our intuition about the model and provide insights about its application.

5 Final considerations

So far, we have not been able to ﬁnd a physical structure that allows

us to change the order of the original derivatives, even if the units are

40 N. Zeraick Monteiro and S. R. Mazorche

adjusted and the new orders are the same in all compartments. Therefore,

we seek the possibility of physically modeling a system, with a formalism

similar to that of Kermack and McKendrick, the creators of the SIR model.

In this sense, we have been discussing the almost unknown model of [6].

Here, we propagate the technique to work with equilibria in this type

of model and present important considerations, not previously explored:

the model is nonlocal, presents a diﬃcult regarding the supposed relation

between the course of the epidemic and the reproduction number and, ﬁ-

nally, there is an unexpected behavior at the initial point. Particularly,

start the simulation of an epidemic in its beginning or be able to say about

the past is a trick question. We aim that the deeper study of the equilib-

rium points and trajectories are fundamental predicting important features

of the model’s application. By other hand, the mathematics of the Frac-

tional Calculus’ models is still a black box of surprises that the assembling

between analytical and numerical studies can help to investigate.

Future works intend to close some results not fully explored, such as the

stability of equilibrium points in the general case and the use of an extrinsic

infectivity arising from Mittag-Leﬄer functions. The proposal of a variable

extrinsic infectivity was ﬁrst illustrated in [4] and made it possible to ﬁne-

tune the COVID-19 data, which leads us to new opportunities.

Acknowledgements

This study was ﬁnanced in part by the Coordenação de Aperfeiçoamento de

Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. We also thank

the team of the National Meeting of Mathematical Analysis and Applications

(ENAMA 2021), for the invitation to submit.

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